284 WITHDRAWN FROM THE ACTION OF GRAVITY. 



height of the frame, and until then they occupy, under a perceptibly plain form, 

 the lateral faces of the prism. 



In the hexagonal frame of the preceding paragraph, the distance between the 

 points where two liquid edges proceeding from one of the summits of the central 

 film attach themselves to the same lateral solid edge is about double the diame- 

 ter of the inscribed circle. In the octagonal frame, as we have just seen, it is 

 somewhat smaller than the half of that diameter ; in a heptagonal frame it is, 

 as might be expected, intermediate between those two values, and equal, nearly, 

 to three-fourths of the same diameter. I have tried, also, a decagonal frame, 

 and in this the distance in question is but the sixth of the diameter. 



The facts just recited would constitute an exception to our law, according to 

 which one film ought to rest on each of the solid edges of the frame, since, reck- 

 oning from either base to the points at which the oblique liquid edges arise, two 

 films are attached to each of the lateral solid edges ; but, as I have shown, the 

 laminar systems in question are imperfect systems. These facts are not owing 

 to the thickness of the metallic threads : they still present themselves when the 

 lateral edges of the frames are of the finest iron wire ; only, in this case, the 

 separation of the points of attachment of the oblique liquid edges is a little 

 greater. 



§ 33. Having afterwards employed an octagonal frame of which the height 

 was a third of the diameter of the inscribed circle, I found that although, accord- 

 ing to the value before given for the separation of these points of attachment, 

 all of them should have been at the summits of the prism, it was not so : the 

 points in question were still at a certain distance from the summits, and their 

 separation was not more than the sixth of the diameter of the inscribed circle ; 

 hence the octagonal film was still enlarged. The same effect was produced with 

 a heptagonal prism of which the height was half the diameter of the inscribed 

 circle — that is to say, less than the separation of the points of attachment as pre- 

 viously estimated in regard to prisms of that number of sides. The same thing 

 occurs in the hexagonal prism, since with a frame of this kind, (§ 26,) the height 

 of which was but lA- of the diameter of the inscribed circle, the points of attach- 

 ment of the oblique liquid edges were still found to be a small distance from the 

 summits. 



§ 34. In order to discover the cause of these last facts, let us consider an 

 octagonal or heptagonal prismatic frame sufficiently high for the films which 

 proceed from the sides of the bases to occupy under a plane form considerable 

 portions of the lateral faces. At the places where these films quit the faces in 

 question to direct themselves towards the sides of the central polygonal film, 

 they are necessarily convex towards the exterior in the direction of their length ; 

 but in consequence of the necessity of a mean curvature null, it is requisite that 

 at these same places they should be concave towards the exterior in the direction 

 of their breadth. If, then, we conceive the frame to be traversed by two planes 

 perpendicular to its axis and passing by the two series of points where arise, on 

 the lateral solid edges, the oblique liquid edges, these two planes will cut the 

 films by arcs concave towards the exterior, and if Ave imagine these arcs solidi- 

 fied, the equilibrium of the system will not be disturbed. According to this, if 

 we constructed a frame having for its height the separation of the points of 

 attachment of the oblique liquid edges on the same lateral solid edge, and gave 

 to the wires which form the sides of the bases the curvature of the above arcs, 

 it is clear that the laminar system realized in this frame would have its oblique 

 liquid edges proceeding exactly from the summits ; but with a frame of th-tt 

 height or of a less height and the sides of whose bases are straight, the condi- 

 tion relative to the transverse curvatures of the re-entering films, and conse- 

 quently to the form of equilibrium of these films, can evidently not be sati?-fied 

 unless the points of attachment of the oblique liquid edges hi placed at a certam 

 distance from the summits on the lateral solid edges. 



